Buckingham function

The different r " terms in the modified function are not linearly independent and, therefore, inclusion of additional terms does not guarantee the improvement in the fit. Buckingham function also suffers from the fact that as r —> 0,... 

The equations can be simplified by reducing the number of parameters required. This is demonstrated for the example of the Buckingham function (Eq. 2.28),... 

A typical set of nonbonded potential functions (Scott and Scheraga, 1966c Ooi et al., 1967) obtained by the procedure described above is given in Table 16. Those of Brant and Flory (1965c) and of Brant et al. (1967), obtained by essentially the same procedure, differ somewhat from these because of the different values selected for the parameters of Table 15. Liquori (1966) has used a combination of Lennard-Jones and Buckingham functions, based primarily on work of Mason and Kreevoy (1955). 

In both equations, d is the separation between the atoms. The Lennard-Jones potential is simpler and computationally less demanding and is therefore favored for models of macromolecules such as proteins and DNA. The Buckingham function more closely resembles the energy relationship and is preferred when higher accuracy is required. The latter function is available in MOMEC and we will concentrate on this. 

Both types of functions are commonly used. Several sets of a, b, c, and d, coefficients are available [1-3]. Equally good results can be obtained using Lennard-Jones-type functions alone or Buckingham-type functions alone or mixtures of Lennard-Jones and Buckingham functions [4]. The attraction coefficients in these expressions are generally but not always calculated with the formula of Slater and Kirkwood [5] ... 

The non-dimensional parameter 77 can now be determined from eqn. (k) and then introduced into the Buckingham function... 

Kitaigorodsky gives non-bonded interaction parameters for use with Buckingham functions on aromatic systems. He also gives a rather special potential energy function of the Buckingham type for peptide work, a "universal potential with just one para-meter for each Interaction type. This set has been... 

Laier and Larsen used three-parameter Buckingham functions com-posed from Allinger s two-parameter Lennard-Jones functions . A quite unnecessary complication, as the program would work equally well with Lennard-Jones functions. 

If computing time does not play the major role that it did in the early 1980s, the [12-6] Lennard-Jones potential is substituted by a variety of alternatives meant to represent the real situation much better. MM3 and MM4 use a so-called Buckingham potential (Eq. (28)), where the repulsive part is substituted by an exponential function ... 

The range of systems that have been studied by force field methods is extremely varied. Some force fields liave been developed to study just one atomic or molecular sp>ecies under a wider range of conditions. For example, the chlorine model of Rodger, Stone and TUdesley [Rodger et al 1988] can be used to study the solid, liquid and gaseous phases. This is an anisotropic site model, in which the interaction between a pair of sites on two molecules dep>ends not only upon the separation between the sites (as in an isotropic model such as the Lennard-Jones model) but also upon the orientation of the site-site vector with resp>ect to the bond vectors of the two molecules. The model includes an electrostatic component which contciins dipwle-dipole, dipole-quadrupole and quadrupole-quadrupole terms, and the van der Waals contribution is modelled using a Buckingham-like function. 

From electronic structure theory it is known that the repulsion is due to overlap of the electronic wave functions, and furthermore that the electron density falls off approximately exponentially with the distance from the nucleus (the exact wave function for the hydrogen atom is an exponential function). There is therefore some justification for choosing the repulsive part as an exponential function. The general form of the Exponential - R Ey w function, also known as a ""Buckingham " or ""Hill" type potential is... 

The pressure dependence of wavenumbers has been investigated theoretically by LD methods on the basis of a Buckingham 6-exp potential. In the studies of Pawley and Mika [140] and Dows [111] the molecules were treated as rigid bodies in order to obtain the external modes as a function of pressure. Kurittu also studied the external and internal modes [141] using his deformable molecule model [116]. The force constants of the intramolecular potential (modified UBFF) were obtained by fitting to the experimental wavenumbers. The results of these studies are in qualitative agreement with the experimental findings. 

The same dimensionless functionality is apparent as from the Buckingham pi method. 

At short range, a plot of In V versus r will be a straight line with a slope (equal to a) and an intercept (equal to A) and thus the value of A and a for Buckingham potential can be determined. The point where the potential V becomes zero would fix the value of B (when n = 6 or any other fixed value). The function can be made more flexible by modifying as... 

The first three terms, stretch, bend and torsion, are common to most force fields although their explicit form may vary. The nonbonded terms may be further divided into contributions from Van der Waals (VdW), electrostatic and hydrogen-bond interactions. Most force fields include potential functions for the first two interaction types (Lennard-Jones type or Buckingham type functions for VdW interactions and charge-charge or dipole-dipole terms for the electrostatic interactions). Explicit hydrogen-bond functions are less common and such interactions are often modeled by the VdW expression with special parameters for the atoms which participate in the hydrogen bond (see below). 

Buckingham TA, Candinas R, Attenhofer C, et al. Systolic and diastolic function with alternate and combined... 

The Lennard-Jones (6-12) potential has served very well as an inter-molecular potential and has been widely used for statistical mechanics and kinetic-theory calculations. It suffers, however, from having only two adjustable constants, and there is no reason why it should not gradually be replaced by more flexible and more realistic functions. Recently a number of applications have been made of the Buckingham (6-exp) potential [Eq. (82)], which has three adjustable parameters. For this potential the first approximation to the coefficient of diffusion is written by Mason (M3) in the form... 

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